Basic topology question

**Drexel28** · January 31st 2010, 08:54 PM

This is relatively easy, but it requires an astute observation. You use this quite a bit later in topology so I guess it's a good challenge question.

NOTE: If you have already seen the solution to this, please don't ruin the fun! This is aimed at people who haven't seen a lot of topology before.

Problem: Define $\mathbb{R}^{\infty}$ to be the set of all square-summable real sequences $\{x_n\}$ equipped with the metric $d\left(\{x_n\},\{y_n\}\right)=\sqrt{\sum_{n=1}^{\i nfty}\left|x_n-y_n\right|^2}$ . Prove that this metric space is separable.

Note: Separable means it contains a countable dense subset. Also, while my book calls this $\mathbb{R}^{\infty}$ it is most commonly known as $\ell^2$ .

**Drexel28** · February 2nd 2010, 11:30 PM

Originally Posted by Drexel28

This is relatively easy, but it requires an astute observation. You use this quite a bit later in topology so I guess it's a good challenge question.

NOTE: If you have already seen the solution to this, please don't ruin the fun! This is aimed at people who haven't seen a lot of topology before.

Problem: Define $\mathbb{R}^{\infty}$ to be the set of all square-summable real sequences $\{x_n\}$ equipped with the metric $d\left(\{x_n\},\{y_n\}\right)=\sqrt{\sum_{n=1}^{\i nfty}\left|x_n-y_n\right|^2}$ . Prove that this metric space is separable.

Note: Separable means it contains a countable dense subset. Also, while my book calls this $\mathbb{R}^{\infty}$ it is most commonly known as $\ell^2$ .

Here is the answer. Try to prove it.

Spoiler: